Question

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Answer 1

Step-by-step explanation:

Time taken by pipe A to fill the cistern = t1.

Time taken by pipe B to fill the cistern = t2.

So if both are open can fill the tank in 3+1/13 (=40/13)minutes,

So ( 1/t1 + 1/t2 )*( 40/13 ) = 1

1/t1 + 1/t2 = 13/40. — (1)

As it's given that one tank take 3 minutes more then other if individually opens.

t1 - t2 = 3

From equation (1)

1/t1 + 1/(t1 - 3) = 13/40

We get t1 = 8 minutes

And t2 = 5 minutes.

Thus pipe A requires 8 minutes and pipe B requires 5 minutes to fill the cistern.

Answer 2

Let the Capacity of the tank to be filled be : T

Let the time taken by the Bigger tap to fill the Tank be : P min

$\implies \mathsf{In\;One \;Minute\; the\; Bigger\; Tap \;fills \;the \;Tank \;by : \dfrac{T}{P}}$

Given : Smaller tap takes 3 minutes more than bigger tap to fill the tank

It means : Time taken by the smaller tap to fill the tank = (P + 3) min

$\implies \mathsf{In\;One \;Minute\; the\; Smaller\; Tap \;fills \;the \;Tank \;by : \dfrac{T}{P + 3}}$

When two taps are opened together at same time :

In One Minute the two taps fill the Tank by :

$\mathsf{\implies \bigg(\dfrac{T}{P} + \dfrac{T}{P + 3}\bigg)}$

$\mathsf{Given : Two\; taps \;fill \;the \;tank \;completely \;in \;\dfrac{40}{13} \;minutes}$

$\mathsf{\implies \dfrac{40}{13}\bigg(\dfrac{T}{P} + \dfrac{T}{P + 3}\bigg) = T}$

$\mathsf{\implies \dfrac{40}{13}\bigg(\dfrac{1}{P} + \dfrac{1}{P + 3}\bigg) = 1}$

$\mathsf{\implies \dfrac{40}{13}\bigg(\dfrac{P + 3 + P}{P(P + 3)}\bigg) = 1}$

$\mathsf{\implies 40(2P + 3) = 13(P^2 + 3P)}$

$\mathsf{\implies 13P^2 + 39P = 80P + 120}$

$\implies \mathsf{13P^2 - 41P - 120 = 0}$

$\implies \mathsf{13P^2 - 65P + 24P - 120 = 0}$

$\implies \mathsf{13P(P - 5) + 24(P - 5) = 0}$

$\implies \mathsf{(P - 5)(13P + 24) = 0}$

$\mathsf{\implies P = 5\;\;\;(or)\;\;\;P = \dfrac{-24}{13}}$

$\mathsf{As \;Time \;cannot\; be \;Negative\; \implies \;P \neq \dfrac{-24}{13}}$

$\mathsf{\implies P = 5}$

$\implies \textsf{Time taken by Bigger tap to fill the Tank = 5 minutes}$

$\implies \textsf{Time taken by Smaller tap to fill the Tank = (5 + 3) = 8 minutes}$